Symmetry, Integrability and Geometry: Methods and Applications
SIGMA 9 (2013), 052, 23 pages
Invariant Discretization Schemes
Using Evolution–Projection Techniques?
Alexander BIHLO
†‡
and Jean-Christophe NAVE
‡
†
Centre de recherches mathématiques, Université de Montréal,
C.P. 6128, succ. Centre-ville, Montréal (QC) H3C 3J7, Canada
E-mail: bihlo@crm.umontreal.ca
‡
Department of Mathematics and Statistics, McGill University,
805 Sherbrooke W., Montréal (QC) H3A 2K6, Canada
E-mail: jcnave@math.mcgill.ca
Received September 27, 2012, in final form July 28, 2013; Published online August 01, 2013
http://dx.doi.org/10.3842/SIGMA.2013.052
Abstract. Finite difference discretization schemes preserving a subgroup of the maximal Lie
invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for non-invariant schemes
of the heat equation in computational coordinates. We propose a new methodology for handling moving discretization grids which are generally indispensable for invariant numerical
schemes. The idea is to use the invariant grid equation, which determines the locations of
the grid point at the next time level only for a single integration step and then to project the
obtained solution to the regular grid using invariant interpolation schemes. This guarantees
that the scheme is invariant and allows one to work on the simpler stationary grids. The
discretization errors of the invariant schemes are established and their convergence rates are
estimated. Numerical tests are carried out to shed some light on the numerical properties
of invariant discretization schemes using the proposed evolution–projection strategy.
Key words: invariant numerical schemes; moving frame; evolution–projection method; heat
equation
2010 Mathematics Subject Classification: 65M06; 58J70; 35K05
1
Introduction
Discretization schemes for differential equations that are not solely constructed for the sake of
reducing the local discretization error as much as possible, but rather to preserve some of the
intrinsic properties of these differential equations have become increasingly popular over the last
decades. While preserving one of these properties, namely conservation laws, led to the design
of conservative discretization schemes which are quite popular in the scientific community [4,
19, 24] (and in particular in the geosciences, e.g. [18, 31]), there are other geometric features
of differential equations that can be attempted to be preserved as well that have received less
attention from the side of numerical analysis so far. One of these features are symmetries of
differential equations. While there have been theoretical advancements on the methodologies
of finding numerical schemes that preserve the maximal Lie invariance groups of systems of
differential equations over the past 20 years or so [13, 22, 30, 35], little is known about the
numerical properties of these invariant schemes. A part of the problem is that while conservation
laws are always properties of the solutions of a differential equation, symmetries are by definition
properties of differential equations. Therefore, it is a standing question whether a discretization
?
This paper is a contribution to the Special Issue “Symmetries of Differential Equations: Frames, Invariants
and Applications”. The full collection is available at http://www.emis.de/journals/SIGMA/SDE2012.html
2
A. Bihlo and J.-C. Nave
scheme that preserves numerically a property of a differential equation also improves the quality
of the numerical solution of that discretized differential equation.
The present paper is devoted to an investigation of this question and related problems
exemplified with invariant discretization schemes for the linear heat equation. The heat equation
is particularly suited for this investigation as it is a canonical example in the group analysis of
differential and difference equations. Moreover, there are already several studies devoted to invariant numerical schemes for this equation [1, 14, 35]. At the same time, in none of these existing
works a deeper background analysis of the numerical properties (e.g. order of approximation or
stability) of the developed schemes was investigated. A first account on numerical properties
of invariant numerical schemes for the heat equation was given in [12], in which a numerical
comparison of invariant and non-invariant schemes for the heat equation regarding accuracy
was presented.
There are several reasons why less attention has been paid so far on the numerical analysis of
invariant schemes (with the exception of the works [10, 12]). One of the reasons is that the field
of invariant discretization schemes is still in its early stages, with new conceptual algorithms
being developed only recently [2, 5, 6, 13, 21, 22, 25, 30]. Another reason is that invariant finite
difference schemes generally require the use of adaptive moving meshes, i.e. it is necessary to
include a non-trivial mesh equation in the discretization problem. Moving meshes lead to nonuniform grid point distributions and, in the multi-dimensional case, to non-orthogonal grids.
The analysis of schemes on such meshes is considerably more difficult than that for related
difference schemes on fixed, uniform and orthogonal meshes. Due to this second reason, most
invariant numerical schemes so far have been constructed only for (1+1)-dimensional single
evolution equations, as in that case moving meshes can be handled still with limited effort.
Although we will be concerned with a (1+1)-dimensional equation in the present paper too, the
methods used subsequently can be employed in the multi-dimensional case without substantial
modification.
The new approach we propose here is to use the invariant grid equation only for a single time
step and then to interpolate the solution to the regular grid. The important observation is that
this interpolation can be done in an invariant way, i.e. projecting the solution does not break
the invariance of the scheme itself. At the same time, the possibility to project the solution of
an invariant scheme to a regular grid is highly desirable as in the multi-dimensional case a freely
evolving grid can cause severe numerical problems. Moreover, for realistic numerical models, as
e.g. employed in weather and climate predictions, it is in general hard to use adaptive meshes
as the discretization of the governing equations is only one part of such model. Other parts are
related to the numerical data assimilation, i.e. the preparation of the initial conditions for the
numerical model and this step usually involves the forecasting model itself. As the assimilation
of the initial conditions cannot be done on an evolving mesh (because the data are given at fixed
locations only) this at once renders invariant schemes on moving meshes impractical. Equally
important, any realistic numerical model for a nonlinear system of partial differential equations
has to contain subgrid-scale models, which mimic the effects of processes taking place at those
scales that the numerical model is not capable of resolving explicitly [33, 34]. The construction
of subgrid-scale models for non-resolved physical processes involves in general ad-hoc arguments
and these arguments rely on the particular scale on which the unresolved processes take place.
As a moving mesh locally changes the resolution and thus impacts what processes are explicitly
resolved by a numerical model, subgrid-scale schemes have to be designed that can operate on
grids with varying resolution. For realistic processes (which are usually not self-similar), this
might be difficult to achieve in practice.
All what was said above objects against invariant numerical schemes for multi-dimensional
systems of differential equations using freely evolving meshes. Thus, whether mathematically
feasible or not, such schemes would be of less practical interest. This is why other approaches
Invariant Discretization Schemes Using Evolution–Projection Techniques
3
should be sought that on the one hand allow one to retain the invariance group of a system
of differential equations in the course of discretization and on the other hand yield schemes
that are practical to avoid the above mentioned and related problems. The proposed invariant
evolution–projection strategy we are going to introduce below may be considered as one such
approach.
The further organization of this article is as follows. The subsequent Section 2 features a summary and some extensions on the various methods to construct invariant discretization schemes.
In Section 3 the heat equation along with its maximal Lie invariance group G is presented. It
is discussed which subgroup G1 of G we aim to present when constructing invariant numerical
discretization schemes. The selection of G1 is based on preserving the class of periodic boundary value problems we are focussing on. Section 4 contains the construction of an equivariant
moving frame for G1 along with a presentation of some lower order differential invariants of G1 .
In Section 5 invariant discretization schemes for the heat equation in computational coordinates
are found. The local discretization errors of these schemes are established in Section 6. In Section 7 we introduce the new idea of invariant interpolation schemes that can be used to project
the numerical solution obtained from an invariant scheme on a moving mesh to the regular grid.
The numerical analysis as well as some numerical tests for the schemes proposed in this paper
are found in Section 8. The summary of this article is presented in the final Section 9.
2
Construction of invariant discretization schemes
The construction of invariant discretization schemes for differential equations can be seen as
a part of the ongoing effort to turn group analysis into an efficient tool for the analysis of
difference equations, see e.g. the review article [25]. As of now, there are three main methods
that are used to construct invariant discretization schemes.
2.1
Dif ference invariant method
The first method was developed by V. Dorodnitsyn, see [1, 13, 14, 25, 35]. It uses the infinitesimal generators of one-parameter symmetry groups of the system of differential equations under
consideration that span the maximal Lie invariance algebra g of this system. These generators
are of the form
v = ζ j (x, u)∂xj + η α (x, u)∂uα = ζ(x, u)∂x + η(x, u)∂u ,
where x = (x1 , . . . , xp ) and u = (u1 , . . . , uq ) are the tuples of independent and dependent
variables, respectively. Here and in the following, the summation convention over repeated
indices is used. Rather than prolonging v to higher order derivatives of u with respect to x,
which is standard in the symmetry analysis of differential equations [3, 26, 29], in this method
the vector fields are prolonged to all the points of the discretization stencil, i.e. the collection
of grid points which are necessary to approximate a given system of differential equation up to
a desired order. This prolongation is of the form
pr v =
m
X
ζ(xi , ui )∂xi + η(xi , ui )∂ui ,
i=1
where xi = (x1i , . . . , xpi ) and ui = (u1i , . . . , uqi ), i.e. it is done by evaluating the vector field v at
all m stencil points zi = (xi , ui ) and summing up the result. An example for such a prolongation
is given in Remark 1 in Section 5.
As a next step, the invariants of the group action are found by invoking the infinitesimal
invariance criterion [13, 26], which in the present case is pr v(I) = 0. The functions I that fulfill
this condition for all v ∈ g are termed difference invariants.
4
A. Bihlo and J.-C. Nave
Once the difference invariants on the stencil space are found, one then has to assemble
these invariants together to a finite difference approximation of the given system of differential
equations. By construction, this procedure guarantees that the resulting numerical scheme is
invariant under the symmetry group of the original system of differential equations.
The main drawback of this method is that it might be hard to find a combination of difference
invariants that approximates a system of differential equations in the multi-dimensional case.
The problem is, as discussed in the introduction, that invariant schemes generally require the
use of moving and/or non-orthogonal grids. Formulating consistent discretization schemes using
difference invariants as building blocks on moving meshes is rather challenging in higher dimensions and thus limited the application of this method to the case of single (1 + 1)-dimensional
evolution equations. We stress though that this problem only enters at the stage of combining difference invariants to a discretization scheme. Computing difference invariants in the
multi-dimensional case can be done as effectively as computing differential invariants for multidimensional problems using infinitesimal techniques.
2.2
Invariant moving mesh method
Retaining the invariance of finite difference schemes under the maximal Lie invariance groups
of physically relevant time-dependent differential equations often requires the use of moving
meshes. This is true both for the finite difference method discussed in the previous Section 2.1
and the moving frame method to be discussed in the next Section 2.3. This kind of mesh
adaptation in which the number of grid points remains constant throughout the integration is
referred to as r-adaptivity in the field of adaptive numerical schemes [7, 20].
The standard strategy to handle r-adaptive meshes is to regard the grid adaptation as a timedependent mapping from a fixed reference space of computational coordinates to the physical
space of the independent variables of the differential equation, i.e. x = x(ξ) for ξ = (ξ 1 , . . . , ξ p )
being the computational variables. Without loss of generality, we assume that ξ 1 = τ = t is the
time variable. The dependent variables u are expressed in the computational space by setting
ū(ξ) = u(x(ξ)). For the sake of simplicity we will omit the bars henceforth.
The significance of the computational coordinates is to provide a reference frame that remains
stationary and orthogonal even in the presence of grid adaptation in the physical space of coordinates. In the course of discretization the variable ξ labels the position of the grid points in the
mesh and this labeling stays unchanged during the mesh adaptation. Thus, the computational
variables can be interpreted physically as Lagrangian coordinates and their invariance under the
motion of the grid is equivalent to the identity of fluid particles in ideal hydrodynamics.
Because by construction the grid remains orthogonal in the ξ-coordinates, the usual finite
difference approximations for derivatives can be used in the space of computational variables.
This simplifies both the practical implementation of the discretization method and the numerical
analysis of the resulting schemes.
The expression of the initial physical system of differential equations in terms of computational variables leads to a system of equations that explicitly includes the mesh velocity xτ ,
which is yet to be determined in order to close the resulting numerical scheme. A prominent
strategy for determining the location of the grid points at the subsequent time level in the onedimensional case uses the equidistribution principle, which in its differential form is (ρxξ )ξ = 0,
where ρ is a monitor function that determines the areas of grid convergence and divergence. For
higher-dimensional problems, equidistribution has to be combined with heuristic arguments,
see [20] for more details.
The invariance of the initial differential equations is brought into the scheme by adequately
specifying the monitor function ρ. In [6] it was pointed out that using monitor functions that
preserving the scale-invariance of a differential equation is particularly relevant in cases where
Invariant Discretization Schemes Using Evolution–Projection Techniques
5
the equation is capable of developing a blow-up solution in finite time, see also [7, 8, 20]. This
finding is generalized upon requiring that the monitor function is chosen in a manner such
that the equidistribution principle is invariant under the same symmetry group as the original
differential equation. For a number of symmetry groups this appears to be possible, see [2] for
an example.
The invariant moving mesh method was recently extended in [2]. The idea of this extension is
to transform the initial system of differential equations to the space of computational coordinates
and to determine the form of the symmetry transformations in the computational space. The
equations in the computational space are then discretized such that the resulting scheme mimics
the transformation behavior of the continuous case. The main advantage of this approach is
that it allows one to retain an initial conserved form of the system of differential equations
and thus to numerically preserve certain conservation laws in the invariant scheme. This is
relevant as preserving conservation laws in the course of invariant numerical modeling is yet
a pristine problem. An exception to this is the discretization of equations that follow from
variational principles, which, if done in a proper way, can lead to the simultaneous preservation
of both symmetries and associated conservation laws, owing to the discrete Noether theorem.
See, e.g. [5] for an example of such an invariant Lagrangian discretization.
Another advantage of the extension proposed in [2] is that it allows one to find invariant
numerical schemes without the detour of difference invariants. This is essential as it can happen
that the single equations in a system of differential equations cannot be approximated directly
in terms on differential invariants but only in combination with other equations of that system.
If this is the case it is not natural to attempt to discretize the system using difference invariants
as this would lead to rather cumbersome discretization schemes.
2.3
Moving frame method
The third method is the most recent one [10, 11, 21, 22, 23, 30]. It relies on the notion of
equivariant moving frames and their important property to provide a mapping that allows one
to associate an invariant function to any given function. As we will mostly work with this
method in the present paper, we describe it in greater detail here. We collect some important
notions on moving frames below, a more comprehensive exposition can be found in the original
references [9, 15, 16, 27, 28, 30].
Definition 1. Let G be a finite-dimensional Lie group acting on a manifold M . A (right)
moving frame ρ is a smooth map ρ : M → G satisfying the equivariance property
ρ(g · z) = ρ(z)g −1 ,
(1)
for all z ∈ M and g ∈ G.
Theorem 1. A moving frame exists in the neighborhood of a point z ∈ M if and only if the
group G acts freely and regularly near z.
Local freeness of a group action means that z̃ = g · z = z for all z from a sufficiently small
neighborhood of each point on M only holds for g being the identity transformation, which
implies that all the group orbits have the same dimension. Here and throughout the paper,
a tilde over a variable denotes the corresponding transformed form of that variable. Regularity
of a group action requires that there exists a neighborhood for each point z ∈ M , which is
intersected by the orbits of G into a pathwise connected subset.
When a group G does not act freely on M , its action can be made free upon extending it to
a suitably high-order jet space J n = J n (M, p) of M , 0 ≤ n ≤ ∞. Locally, the nth order jet space
of a p-dimensional submanifold of M has coordinates z (n) = (x, u(n) ), where as in the previous
6
A. Bihlo and J.-C. Nave
subsections x = (x1 , . . . , xp ) are considered as the independent variables, u = (u1 , . . . , uq ),
q = dim M − p, are the dependent variables and u(n) collects all the derivatives of u with respect
to x of order not greater than n including u as the zeroth order derivatives. In practice, the
prolongation of the group action of G on J n is implemented using the chain rule.
Moving frames are determined using a normalization procedure. The steps to find a moving
frame for a group action G are the following: (i) Define a cross-section to the group orbits.
A cross-section C is any submanifold C ⊂ M of complementary dimension to the dimension r of
the group orbits, i.e. dim C = dim M −r, that intersects each group orbit once and transversally.
Usually coordinate cross-sections are chosen in which some of the coordinates of M (or of J n
if the group action is not free on M ) are set to constants, i.e. z 1 = c1 , . . . , z r = cr . (ii) The
algebraic system z̃ 1 = (g · z)1 = c1 , . . . , z̃ r = (g · z)r = cr is solved for the group parameters
g = (g1 , . . . , gr ). The resulting expression g = ρ(z) is the moving frame.
Moving frames can be used to map any given function to an invariant function by a procedure
called invariantization.
Definition 2. The invariantization of a real-valued function f : M → R using the (right) moving
frame ρ is the function ι(f ), which is defined as ι(f )(z) = f (g · z)|g=ρ(z) = f (ρ(z) · z).
That the function ι(f ) constructed in this way is indeed invariant follows from the equivariance property (1) of the moving frame ρ,
ι(f )(g · z) = f (ρ(g · z)g · z) = f ρ(z)g −1 g · z = f (ρ(z) · z) = ι(f )(z),
which boils down to the definition of an invariant function I, i.e. I(g · z) = I(z). In practice,
a function f (z) is invariantized by first transforming its argument using the transformations
from G and then substituting the moving frame for the group parameters. By definition, an
invariant that is defined on the jet space J n is a differential invariant.
Moving frames can also be constructed on a discrete space. In a finite difference approximation, derivatives of functions are approximated using a finite set of values of these functions, and
all the points needed to approximate the derivatives arising in a system of differential equations
are the points of the stencil introduced in Section 2.1. Because most of the interesting symmetries of differential equations that are broken in standard numerical schemes require the use of
non-orthogonal discretization meshes, it is beneficial to both regard x and u as the dependent
variables and the computational variables ξ as the independent variables as was discussed in the
previous Section 2.2.
Sampling the tuples from the extended computational space Mξ = {(ξ, z)} at discrete points,
i.e. at (ξi , z(ξi )) = (ξi , zi ), one can introduce the space Mξn = {(w1 , . . . , wn ) | ξi 6= ξj for all
i 6= j}, where wi = (ξi , zi ), which can be identified as the joint product space of stencil variables.
Because the identifier ξi of the point wi is required to be unique, each element of Mξn only
includes distinct grid points in the physical space of equation variables. The dimension of the
space Mξn depends on the number of independent and dependent variables in the system of
differential equations and the desired order of accuracy of the approximated derivatives.
It is possible to carry out the construction of the moving frame on Mξn , i.e. to define the
n
n
moving frame by an equivariant mapping ρn
ξ : Mξ → G, where G acts on Mξ by the product
action, g · (w1 , . . . , wn ) = (g · w1 , . . . , g · wn ). Note that the extension of the group action to
the computational variables ξ is trivial, i.e. they remain unaffected by G, ξ˜ = g · ξ = ξ, see [2].
The compatibility between the moving frame ρn
ξ and the moving frame ρ on the space M (or
n
n
an appropriate jet space J ), i.e. that ρξ → ρ in the continuous limit is assured provided that
the cross-section defining the moving frame ρn
ξ in the continuous limit converges to the crosssection defining the moving frame ρ. Once the moving frame is constructed on the discrete
space Mξn of stencil variables, it can be used to invariantize any numerical scheme expressed
Invariant Discretization Schemes Using Evolution–Projection Techniques
7
in computational coordinates. This will be explicitly shown in Sections 5 and 6 where we will
construct invariant schemes for the heat equation.
It is essential that the construction of the moving frame on the grid point space is carried out
in terms of computational coordinates rather than physical coordinates. This can be illustrated
by the following simple example.
Example 1. The Laplace equation uxx +uyy = 0 is, inter alia, invariant under the one-parameter
group of rotations SO(2), x̃ = x cos ε − y sin ε, ỹ = x sin ε + y cos ε. Let us obtain the moving
frame ρ for this group action from the normalization condition ux = 0, i.e. we determine the
moving frame on the first jet space J 1 (M, 2), ρ = ρ(x, y, u, ux , uy ). Prolonging the transformations from SO(2) to the derivative ux leads to ũx̃ = ux cos ε − uy sin ε and thus the moving frame
is ε = arctan(ux /uy ).
Let us now find the product frame from the discrete normalization condition udx = 0. Computing udx in the naı̈ve way, udx = (ui+1 − ui−1 )/(xi+1 − xi−1 ), we fail as
ũdx̃ =
ui+1 − ui−1
= 0,
(xi+1 − xi−1 ) cos ε − (yi+1 − yi−1 ) sin ε
cannot be solved for the group parameter ε. On the other hand, setting u = u(x(ξ 1 , ξ 2 ), y(ξ 1 , ξ 2 ))
and expressing udx in terms of the computational variables ξ 1 , ξ 2 , the normalization udx = 0 reads
d ỹ d − ũd ỹ d
d xd sin ε + y d cos ε − ud xd sin ε + y d cos ε
ũ
u
1 ξ2
2 ξ1
1
2
2
2
1
1
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ũdx̃ = d d
=
= 0.
x̃ξ1 ỹξ2 − x̃dξ2 ỹξd1
xdξ1 yξd2 − xdξ2 yξd1
This expression can be solved for ε and it gives
!
d
udξ1 yξd2 − udξ2 xdξ1
ux
= arctan
,
ε = arctan
d
d
d
d
udy
uξ2 yξ1 − uξ1 xξ2
which in the continuous limit goes to ε = arctan(ux /uy ) as required.
3
Lie symmetries of the heat equation
The one-dimensional linear heat transport equation is
ut − uxx = 0,
(2)
where we scaled the thermal diffusivity ν to 1, i.e. equation (2) is in non-dimensional form.
The heat equation (2) admits the following infinitesimal generators of one-parameter groups,
which generate the maximal Lie invariance algebra g of equation (2):
∂t ,
2
∂x ,
u∂u ,
2t∂t + x∂x ,
2t∂x − xu∂u ,
2
α(t, x)∂u ,
4t ∂t + 4tx∂x − (x + 2t)u∂u ,
(3)
where α runs through the set of solutions of equation (2), see e.g. [26]. These vector fields
generate (i) time-translations, (ii) space translations, (iii) scalings in u, (iv) simultaneous scalings
in t and x, (v) Galilean boosts, (vi) inversions and (vii) the superposition principle symmetry.
In this paper, we will construct invariant numerical schemes for a class of initial value problems
of the heat equation using periodic boundary conditions. This class of initial-boundary value
problems only admits a subgroup of the symmetry group of the heat equation as inversions are
no longer admitted; inversions do not send an initial value problem from the considered class to
another initial value problem. The symmetries associated with the first five vector fields in (3)
8
A. Bihlo and J.-C. Nave
are compatible with the class of initial-boundary value problems we are interested in, i.e. they
map the class of initial-boundary value problems for the heat equation under consideration to
itself. This re-interpretation of symmetries of differential equations without initial and boundary
conditions as equivalence transformations for a class of initial-boundary value problems was
recently pointed out in [2].
In what follows we will thus focus our attention on constructing numerical schemes that preserve the symmetries generated by the first five operators of (3). The associated subalgebra of g
will be denoted by g1 . We do not require to preserve the linearity operator here by construction.
At the same time, as will be shown in Section 8 the numerical schemes we propose in this paper
preserve the linearity property up to the discretization error expected.
4
Moving frame and dif ferential invariants for the heat equation
We determine the moving frame for the subgroup G1 of transformations associated with the
subalgebra g1 . Transformations of G1 are of the form
t̃ = e2ε4 (t + ε1 ),
x̃ = eε4 (x + ε2 + 2ε5 t),
2
ũ = eε3 −ε5 x−ε5 t u.
(4)
Because the group action of G1 is not free on M = {(t, x, u)} we construct the moving frame
on a suitably high-order jet space. In the present case, the group action of G1 becomes free on
J 1 = J 1 (M, 2). Thus, it is necessary to extend the transformations (4) to derivatives of u with
respect to t and x.
Using the chain rule we can compute the transformed operators of total differentiation, which
read as
Dt̃ = e−2ε4 (Dt − 2ε5 Dx ),
Dx̃ = e−ε4 Dx ,
where Dx = ∂x + ux ∂u + utx ∂ut + uxx ∂ux + · · · and Dt = ∂t + ut ∂u + utt ∂ut + utx ∂ux + · · · denote
the usual operators of total differentiation. With the transformed total differentiation operators
at hand it is possible to compute the transformed partial derivatives of u with respect to t and x.
The transformation rules for the lowest order derivatives are
2
2
ũt̃ = e−2ε4 +ε3 −ε5 x−ε5 t ut − 2ε5 ux + ε25 u ,
ũx̃ = e−ε4 +ε3 −ε5 x−ε5 t (ux − ε5 u),
2
ũx̃x̃ = e−2ε4 +ε3 −ε5 x−ε5 t uxx − 2ε5 ux + ε25 u .
In fact, for the construction of the moving frame already the knowledge of the first order
derivatives is sufficient.
We compute the moving frame for the five-parameter group of transformations of the form (4)
using the following normalization conditions which determine a valid cross-section to the group
orbits of the prolonged action of G1 on J 1 ,
t = 0,
x = 0,
u = 1,
ut = 1,
ux = 0.
(5)
The moving frame is computed by taking the transformed form of the normalization conditions,
t̃ = 0, x̃ = 0, ũ = 1, ũt̃ = 1 and ũx̃ = 0 and by solving the resulting algebraic system for the group
parameters ε1 , . . . , ε5 . The result of this computation is the following moving frame g = ρ(z (1) ),
ux ux
u2x
ε1 = −t,
ε2 = − x + 2t
,
ε3 = − ln u − x − t 2 ,
u
u
u
(6)
r
2
ut ux
ux
ε4 = ln
− 2,
ε5 =
.
u
u
u
Invariant Discretization Schemes Using Evolution–Projection Techniques
9
With the moving frame at hand, we can invariantize any of the partial derivatives of u with
respect to t and x and thus obtain a complete set of differential invariants for the subgroup G1
of the maximal Lie invariance group of the heat equation. As an example, invariantizing the
derivative uxx , i.e. computing ι(uxx ) as (g · uxx )|g=ρ(z (1) ) we produce the differential invariant
ι(uxx ) =
uu2xx − u2x
.
uut − u2x
Invariantizing the heat equation, i.e. computing ι(ut − uxx ) = 0 and recalling that ι(ut ) = 1, we
obtain
u(ut − uxx )
= 0,
uut − u2x
which yields the original heat equation expressed in terms of differential invariants. This reexpression of a differential equation using the differential invariants of its symmetry group is
known as the replacement theorem [9].
5
Invariant discretization of the heat equation
The invariant discretization of equation (2) cannot be done on a fixed, uniform grid. To see
this, let us check the transformation behavior of the grid equation xn+1
− xni = 0, which is the
i
definition of a stationary grid, under the transformations (4). This yields
n+1
n
ε4
n
n+1
n
x̃n+1
−
x̃
=
e
x
−
x
+
2ε
t
−
t
,
5
i
i
i
i
which is only zero in the case when ε5 = 0. Stated in another way, a discretization on a fixed
grid can at most preserve the symmetry subgroup of G, which is generated by the first four
elements of the maximal Lie invariance algebra g of the heat equation (3).
Thus, the discretization of (2) preserving G1 will require the use of moving grids. For this
reason it is convenient to express (2) in terms of computational coordinates initially, i.e. we set
u(τ, ξ) = u(τ, x(τ, ξ)), where ξ is the single spatial computational variable and τ = t. The heat
equation in this set of coordinates reads
uξ
xξξ
1
uτ − xτ
− 2 uξξ −
uξ = 0.
(7)
xξ
xξ
xξ
So as to find the invariant discretization of the heat equation in the form (7), we determine the
moving frame in the space of stencil variables Mξ4 using the discrete analogs of the normalization
conditions (5) expressed in terms of computational coordinates.
For the sake of convenience we introduce the notation h+ = xni+1 − xni , h− = xni − xni−1 ,
∆τ = τ n+1 − τ n . The discretization stencil we use is depicted in Fig. 1.
The appropriate normalization conditions for a compatible moving frame ρ4
ξ are
τ n = 0,
xni = 0,
uni = 1,
udt = 1,
udx = 0,
udx =
uni+1 − uni−1
h+ + h−
(8)
where
udt =
un − uni−1
un+1
− uni
i
− xdτ i+1
,
∆τ
h+ + h−
are the discretizations of the first time and space derivatives expressed in computational coordinates and xdτ = (xn+1
− xni )/∆τ is the discrete grid velocity. Replacing the single equations
i
10
A. Bihlo and J.-C. Nave
Figure 1. Stencil for an invariant discretization scheme of the heat equation.
in the above normalization conditions by their respective transformed expressions and solving
the resulting algebraic system for the group parameters we obtain the following moving frame
on the space of stencil variables Mξ4 ,
n
xni
n
(ln u)dx
ln uni
ε1 = −τ ,
ε2 = −
+ 2τ
,
ε3 = −
−


d 2
d
d
un+1
− uni
i
1  exp −∆τ xτ (ln u)x + (ln u)x
,
ε4 = ln
2
uni ∆τ
xni (ln u)dx
−τ
n
2
(ln u)dx
,
(9)
ε5 = (ln u)dx ,
where we introduced (ln u)dx = (ln uni+1 − ln uni−1 )/(h+ + h− ). This moving frame is compatible
with the moving frame (6) in that it converges to (6) in the continuous limit ∆ξ → 0 and ∆τ → 0
upon using
h− = xξ ∆ξ − xξξ (∆ξ)2 /2 + O ∆ξ 3 ,
h+ = xξ ∆ξ + xξξ (∆ξ)2 /2 + O ∆ξ 3 ,
(10)
xn+1
= xni + xτ ∆τ + O ∆τ 2 ,
uni+1 = uni + uξ ∆ξ + uξξ (∆ξ)2 /2 + O ∆ξ 3 ,
i
uni−1 = uni − uξ ∆ξ + uξξ (∆ξ)2 /2 − O ∆ξ 3 ,
un+1
= uni + uτ ∆τ + O ∆τ 2 .
i
The moving frame (9) can now be used to invariantize any non-invariant finite difference
discretization of (7) on Mξ4 . To illustrate this, we invariantize the standard FTCS (forward in
time centered in space) scheme
udt −
(h+
4
uni+1 + uni−1 − 2uni − (h+ − h− )udx = 0.
−
2
+h )
This is done by first replacing all terms by their respective transformed expressions and substituting the moving frame (9) for the arising group parameters. The result of this procedure is
the invariant scheme
exp −∆τ xdτ (ln u)dx + ((ln u)dx )2 un+1
− uni
i
S=
∆τ
!
n h− /(h+ +h− )
n −h+ /(h+ +h− )
u
u
(11)
i+1
i+1
4 uni+1
+ uni−1
− 2uni
n
n
ui−1
ui−1
−
= 0.
(h+ + h− )2
Again, it can be checked that the above scheme (11) indeed converges to (7) in the limit of
∆ξ → 0 and ∆τ → 0. This will be shown explicitly in Section 6, where we establish the order
of approximation of (11).
Invariant Discretization Schemes Using Evolution–Projection Techniques
11
So as to complete the scheme (11) it is necessary to determine xn+1
, which is the ingredient
i
missing in (11). There are different ways to determine a grid equation, such as using the
equidistribution principle as outlined in Section 2.2. The problem with this strategy in the
present case is that while it might be beneficial from the numerical point of view, it might not
be easy to obtain an invariant discretization of this principle which does not lead to a fully
coupled equation–grid system. In other words, it can happen that the grid equation includes
values of u at both tn and tn+1 . While this coupling is not a problem in the one-dimensional
case, it can lead to a severe restriction of the applicability for multi-dimensional equations as
solving the coupled equation–grid system might then be too expensive.
A G1 -invariant grid equation that circumvents the aforementioned coupling problem can be
derived from the invariantization of xn+1
. This invariantization yields
i
2∆τ
n+1
n+1
ε4
n
n
n
ι xi
xi − xi + +
=e
ln ui−1 − ln ui+1 ,
h + h−
where we did not explicitly substitute the frame value for ε4 . An appropriate grid is then given
through ι(xn+1
) = 0, or
i
M = xn+1
− xni +
i
2∆τ
ln uni−1 − ln uni+1 = 0.
−
+h
(12)
h+
This grid equation is quite similar to the grid equation
+ n n ui−1
ui+1
h
h−
2∆τ
n+1
n
ln
− + ln
= 0,
xi − xi + +
n
−
−
h +h
h
ui
h
uni
(13)
which was found in [1, 14] using the method of difference invariants. This last grid (13) is not
only invariant under the subgroup G1 but under the whole maximal Lie invariance group G of
the heat equation. In the continuous limit, both equation (12) and (13) converge to
xτ = −
2
(ln u)ξ .
xξ
We have tested all our numerical schemes with both (12) and (13) and found that the resulting
schemes give asymptotically the same numerical results. In fact, as in the evolution–projection
strategy that will be introduced in Section 7 we have h+ = h− = h, equation (12) and (13)
coincide.
Remark 1. While the invariantization algorithm guarantees that the scheme (11) is indeed
invariant under the subgroup G1 of the maximal Lie invariance group of the heat equation,
the invariance can be checked in a straightforward fashion using the infinitesimal invariance
criterion as invoked in the Dorodnitsyn method discussed in Section 2.1. Let us recall that this
criterion states that an invariant I of a group action is annihilated by the associated infinitesimal
generators, i.e. v(I) = 0 for all v ∈ g. Because in the present case, the invariants are defined
on the stencil space with coordinates τ n , ∆τ , xni , xni+1 , xni−1 , xn+1
, uni , uni+1 , uni−1 and un+1
,
i
i
we have to prolong the operators of g accordingly. The prolongations of the first five operators
of (3) to the variables of the stencil are
n
i
i
2τ ∂
n
∂un+1 ,
uni ∂uni + uni+1 ∂uni+1 + uni−1 ∂uni−1 + un+1
i
∂xni + ∂xni+1 + ∂xni−1 + ∂xn+1 ,
∂τ n ,
τn
+ 2∆τ ∂∆τ +
xni ∂xni
+
xni+1 ∂xni+1
+
n
xni−1 ∂xni−1
2τ (∂xni + ∂xni+1 + ∂xni−1 ) + 2(τ + ∆τ )∂xn+1 −
i
− xni−1 uni−1 ∂uni−1 − xn+1
un+1
∂un+1 ,
i
i
i
+
xn+1
∂xn+1 ,
i
i
xni uni ∂uni
− xni+1 uni+1 ∂uni+1
12
A. Bihlo and J.-C. Nave
see [13, 25] for more details. It can be checked that pr v(S) = 0 and pr v(M ) = 0 hold on S = 0
and M = 0 for all the prolonged infinitesimal generators and thus S = 0 is a proper invariant
numerical scheme and M = 0 an invariant grid equation.
Remark 2. The heat equation is a linear partial differential equation in two independent
variables. One might thus consider to set up a grid equation not only for spatial but also for
temporal adaptation. The reason why we refrain from spatial-temporal adaptation here is that
the symmetry group G of the heat equation is compatible with flat time layers, i.e. tni+1 −
tni = 0 is a G-invariant equation. Improving the invariant numerical scheme constructed above
using temporal adaptation would thus not allow one a fair comparison against the original noninvariant FTCS scheme for the heat equation. Moreover, flat time layers are well-agreed with
the physics of the heat transfer problem, which affects all points of the domain simultaneously.
6
Numerical properties of the invariant scheme
In this section we investigate the numerical properties of the scheme (11) and related schemes.
We start our consideration with the estimation of the local truncation error of the scheme. The
study of this question is relevant because so far little is known about the relation between the
order of a non-invariant scheme and its invariantized counterpart.
The discretization of the heat equation in computational coordinates (7) can be formally
represented as
!
d
d
u
x
1
ξ
ξξ
udτ − xdτ d − d 2 udξξ − d udξ = 0,
(14)
xξ
(xξ )
xξ
where in the present case we assume that derivatives are approximated with the aid of a standard
FTCS scheme. More general schemes will be considered after the order of the invariantized FTCS
scheme is established.
Theorem 2. The order of the invariant scheme (11) is the same as the order of the scheme (14),
namely first order in time and second order in space, provided that an Euler forward step and
second order centered differences are used to approximate the time and space derivatives arising
in both the differential equation (14) and the normalization conditions (8).
Proof . Invariantizing the scheme (14) using the normalization conditions (8) leads to
1−
4
d
ι
u
= 0,
ξξ
d 2
(15)
ι xξ
where ι(f )(z) denotes the invariantization of the function f (z). By definition, invariantization
of a function f (z) means to transform the argument z and plug in the moving frame for the
group parameters. In the present case, the transformed form of (15) can be written as
2
1−
4eε3 −ε5 x−ε5 t−2ε4 −ε5 h+ n
−
e
ui+1 + eε5 h uni−1 − 2uni = 0.
2
xdξ
Using the normalization condition uni = 1 we obtain that
2
ũni = 1 = eε3 −ε5 x−ε5 t uni
and thus the last expression can be recast as
e2ε4 uni −
(h+
4
+
−
e−ε5 h uni+1 + eε5 h uni−1 − 2uni = 0.
−
2
+h )
(16)
Invariant Discretization Schemes Using Evolution–Projection Techniques
13
Let us now determine the local discretization error in the parameter ε5 . The respective moving
frame component is
ε5 =
ln uni+1 − ln uni−1
,
h+ + h−
which upon using (10) expands to
ε5 =
1 uξ
+ O ∆ξ 2 .
n
xξ ui
(17)
Substituting ε5 into the second term of equation (16) and expanding the exponential functions
in the same term into Taylor series, we obtain after some rearranging
1
uni+1 + uni−1 − 2uni − ε5 xξ ∆ξ uni+1 − uni−1
e2ε4 uni − 2 2
xξ ∆ξ + O ∆ξ 4
1
1 2 2 2 n
2 n
n
n
4
= 0.
+ xξξ ∆ξ ui+1 + ui−1 + ε2 xξ ∆ξ ui+1 + ui−1 + O ∆ξ
2
2
This can be further simplified to
1
e2ε4 uni − 2
xξ
u2ξ
xξξ
uξ
uξξ − n −
ui
xξ
!
+ O ∆ξ 2 = 0.
(18)
It now remains to expand the first term in equation (18). The moving frame component for ε4
in (9) can be recast as
exp −∆τ
e2ε4 uni =
xn+1
− xni ln uni+1 − ln uni−1
i
+
∆τ
h+ − h−
∆τ
ln uni+1 − ln uni−1
h+ − h−
2 !!
un+1
− uni
i
.
Using xin+1 = xni + xτ ∆τ + O(∆τ 2 ) and un+1
= uni + uτ ∆τ + O(∆τ 2 ) and again expanding the
i
exponential function into a Taylor series, we derive
e2ε4 uni = uτ − xτ
2
uξ
1 uξ
− 2 n + O ∆τ, ∆ξ 2 .
xξ
xξ ui
Plugging this into equation (18) we arrive at
uξ
1
− 2
uτ − xτ
xξ
xξ
xξξ
uξξ −
uξ
xξ
+ O ∆τ, ∆ξ 2 = 0,
which completes the proof of the theorem.
A more general statement is the following one:
Theorem 3. The order of spatial discretization of an invariant finite difference scheme for
the heat equation in computational variables equals the order p ∈ N of the spatial discretization
of the associated non-invariant finite difference scheme provided that centered differences of
order p are used to approximate both the derivatives in the heat equation and in the normalization
conditions (8).
14
A. Bihlo and J.-C. Nave
Proof . In view of the general form (15) of the invariantization of scheme (14), we study the
invariantization of the terms xξ and uξξ .
The invariantization of (xdξ )2 = (xξ + O(∆ξ p ))2 is ι((xdξ )2 ) = e2ε4 (x2ξ + O(∆ξ p )) and is of the
same order p if, as required, the moving frame component ε4 stems from the approximation of
udτ = 1 using pth order accuracy and thus only includes approximations of derivatives with that
accuracy.
Let us now investigate the invariantization of discretizations of uξξ . The general form of
a centered difference approximation of uξξ of even order p is
uξξ
p/2
X
1
=
c2p,j unj + O ∆ξ p ,
2
∆ξ
j=−p/2
where c2p,j = 2c1p,j /j, j ∈ A = {−p/2, . . . , −1, 1, . . . , p/2}, c2p,0 = −2
(−1)j+1 (p/2)!2
,
j(p/2 + j)!(p/2 − j)!
c1p,j =
p/2
P
1/i2 and
i=1
j∈A
and c1p,0 = 0 are the coefficients from the pth order approximation of uξ , i.e.
p/2
1 X 1 n
cp,j uj + O ∆ξ p .
uξ =
∆ξ
j=−p/2
See [17] for a discussion of the algorithm for finding the weights ckp,j in higher-order finite
difference approximations of the kth derivative of u. The invariantization of uξξ is
p/2
X
1
ι(uξξ ) =
c2p,j exp ε3 − ε5 xnj − ε25 τ n unj ,
2
∆ξ
j=−p/2
or, upon using the normalization condition uni = 1,
p/2
1 1 X 2
ι(uξξ ) =
cp,j exp(−ε5 ∆xj )unj ,
∆ξ 2 uni
(19)
j=−p/2
where we expand
∆xj =
xnj
−
xni
=
∞
X
(j∆ξ)k ∂ k x
k=1
k!
∂ξ k
,
uni
=
∞
X
(j∆ξ)l ∂ l u
l=0
l!
∂ξ l
.
Using the expressions for ∆xj and uni , the expression (19) can be expanded and rearranged in
powers of j∆ξ in the form
p/2
∞
1 1 X X
ι(uξξ ) =
(−1)k c2p,j Ak (j∆ξ)k ,
∆ξ 2 uni
j=−p/2 k=0
where
A2 =
1
uξξ − ε5 2xξ uξ + xξξ uni + ε25 x2ξ uni .
2
Invariant Discretization Schemes Using Evolution–Projection Techniques
15
The expressions for Ak , k 6= 2, are not required subsequently. The proof is completed upon
substituting for ε5 the corresponding moving frame component (which is of order p if the normalization udx = 0 is approximated with pth order accuracy) and by noting that


for k ∈ {0, 1, 3, . . . p + 2, 2n}, n ∈ N,
p/2
0
X
2 k
cp,j j = 2
.
for k = 2,


j=−p/2
ck 6= 0 else,
where the precise values of the constants ck follow from evaluating the respective sums.
The scheme (11) is only of first order in time τ = t. To construct a scheme that is second
order in time, we can start with a non-invariant scheme (14) and discretize the time derivative udτ
with second order accuracy, i.e. we set udτ = (un+1
− un−1
)/(2∆τ ), where uin−1 is the value of u
i
i
at the previous time step τ n−1 = τ n − ∆τ . It is now necessary to check whether invariantizing
this leapfrog discretization leads to an invariant scheme that is also second order in time.
Theorem 4. Invariantization of the scheme (14) in which a leapfrog step and second order
centered differences are used to approximate the time and space derivatives, leads to an invariant
scheme that is both second order in time and space provided that the normalization conditions (8)
are approximated using discretizations that are of second order.
Proof . To prove this theorem it is sufficient to establish the order of the first term in equation (18). We proceed in an analog manner as in the proof of Theorem 2, i.e. we discretize the
normalization condition udτ = 1 but now with second order accuracy. This yields
n
ũdτ̃
2 n
eε3 −ε5 xi −ε5 τ
=
2e2ε4 ∆τ
2
2
e−ε5 xτ ∆τ −ε5 ∆τ un+1
− eε5 xτ ∆τ +ε5 ∆τ un−1
= 1.
i
i
Using the normalization condition uni = 1 as before and expanding the exponential functions we
derive
e2ε4 uni =
1
+ O ∆τ 2
un+1
− un−1
− ∆τ ε5 xτ + ε25 un+1
+ un−1
i
i
i
i
2∆τ
and upon noting that un+1
+ un−1
= 2uni + O(∆τ 2 ) we obtain
i
i
e2ε4 uni = uτ − xτ
2
uξ
1 uξ
− 2 n + O ∆τ 2 , ∆ξ 2 ,
xξ
xξ ui
where we have substituted the expression (17) for ε5 . Plugging this result into equation (18)
completes the proof of the theorem.
The actual form of the resulting invariant leapfrog scheme is
− exp ∆τ x̌dτ (ln u)dx + ((ln u)dx )2 un−1
exp −∆τ x̂dτ (ln u)dx + ((ln u)dx )2 un+1
i
i
2∆τ
!
n −h+ /(h+ +h− )
n h− /(h+ +h− )
u
u
i+1
i+1
4 uni+1
+ uni−1
− 2uni
uni−1
uni−1
−
= 0,
(h+ + h− )2
where x̂dτ = (xn+1
− xni )/∆τ and x̌dτ = (xni − xin−1 )/∆τ .
i
Higher order in time schemes can be constructed upon invariantizing multi-stage schemes.
Combining this result with the result established in Theorem 3 we have found the following:
16
A. Bihlo and J.-C. Nave
Corollary 1. Invariantizing a non-invariant finite difference scheme for the heat equation in
computational coordinates preserves the spatial and temporal order of the initial non-invariant
finite difference scheme provided that centered differences are used and the normalization conditions for the moving frame are discretized with the same order as the respective derivatives in
the non-invariant finite difference scheme.
7
Invariant interpolation schemes
A common property of invariant numerical schemes for evolution equations possessing a nontrivial maximal Lie invariance group is that it is not possible to use a fixed, orthogonal discretization mesh. The continuous evolution of the mesh, if not handled properly, can lead to
several undesirable properties, such as an overly strong concentration of grid points in certain
regions and therefore too poor a resolution in other parts of the integration domain. The problem gets worse in the multi-dimensional case where mesh tangling or strongly skewed meshes
can occur. But even if the mesh movement can be managed in an optimal way there are various
physical problems for which continuously adapting grids pose a severe challenge. An example
for this are practically all models that are in operational use in weather and climate prediction.
These models employ sophisticated data assimilation strategies and are coupled to subgrid-scale
parameterizations that aim to mimic the effects of unresolved processes on the grid scale variables. Attempting to make use of data assimilation or parameterization schemes on moving
meshes is not only a technical problem that would cause a significant computational overhead
compared to standard schemes but also a conceptual challenge for it is unclear on how to design
parameterization schemes that can operate on grids with varying resolution. In order to promote the ideas of invariant numerical discretization schemes beyond their application to simple
evolution equations it is thus instructive to study possible ways of overcoming the limitations
imposed by the requirement of using moving meshes.
One straightforward idea is to use invariant schemes on fixed (i.e. non-invariant grids). As
was shown in [30] this can lead to improved numerical solutions compared to non-invariant integrators, while still being excelled by the results that can be obtained using completely invariant
schemes. On the other hand, if moving (invariant) meshes are not tractable for a particular
class of problems, preserving the invariance of a system of differential equations at least for the
discretization of the system itself might be a possible trade-off to take.
Another idea is to use an evolution–projection strategy, which will be proposed in the following. This concept relies on using the invariant scheme for the system of differential equations
together with the invariant mesh equations for a single time step followed by the projection of
the numerical solution back to the regular mesh. A similar strategy has proven successful in
semi-Lagrangian time integration schemes [32].
In the present case, the projection step can be practically realized by using interpolation.
Obviously, any standard interpolation scheme can be used to map the numerical solution un+1
i
defined at xn+1
to the uniformly spaced ξ-grid. This, however, can break the invariance of the
i
numerical scheme as a whole and so the question arises whether it is possible to accomplish the
interpolation step in a symmetry-preserving fashion.
In the following we discuss two possible ways of formulating invariant interpolation schemes,
both of which can be used for finding interpolations that allow the re-mapping of the numerical
solution on a moving mesh to a fixed, Cartesian, equally-spaced grid. These ways are the
invariantization of non-invariant interpolation schemes with moving frames and the construction
of interpolations using difference invariants.
Invariantization of interpolation schemes. The moving frame constructed in the course
of invariantizing a finite difference scheme can also be used to invariantize a certain interpolation
Invariant Discretization Schemes Using Evolution–Projection Techniques
17
method. We exemplify this idea by invariantizing the formula for linear interpolation,
un+1
(y) = un+1
+ y − xn+1
i
i
i
n+1
un+1
i+1 − ui
n+1
xn+1
i+1 − xi
,
where y ∈ [xn+1
, xn+1
i
i+1 ]. The invariantization of this expression using a moving frame associated
1
with G yields the invariant interpolation formula
un+1
(y) = Uin+1 + y − xn+1
i
i
Uin+1
n+1
Ui+1
− Uin+1
n+1
xn+1
i+1 − xi
n+1
= exp (ln ûx̂ )d y − xn+1
ui .
i
,
(20)
Note that we have used a slightly different moving frame for the invarianization as we have
used for invariantizing the finite difference scheme for the heat equation. Specifically, this
moving frame is constructed by replacing the normalization condition udx = 0 with ûdx̂ = 0. The
reason for this is that the moving frame used earlier yielded ε5 = (ln ux )d , i.e. it involves the
solutions of ui at the time step τ n rather than at τ n + ∆τ . Irrespectively of what normalization
is used, both interpolations are invariant. Setting y = ξi in the above interpolation formula
yields un+1
on the regular computational grid. Note that the interpolation (20) is consistent in
i
n+1 n+1
n+1
that ui (xi ) = un+1
and un+1
(xn+1
i
i
i+1 ) = ui+1 .
In a similar manner more sophisticated interpolation schemes can be invariantized. In the following, we will use the invariantization of quadratic interpolation. Usual quadratic interpolation
is based on the expression
n+1
u
(y) =
un+1
i−1 Li−1 (y)
+
un+1
Li (y)
i
+
un+1
i+1 Li+1 (y),
Lj (y) =
i+1
Y
k=i−1
k6=j
y − xn+1
k
, (21)
xn+1
−
xn+1
j
k
n+1
where y ∈ [xn+1
i−1 , xi+1 ] and Lj (y) are the Lagrangian interpolation polynomials. Invariantizing
this formula using the same moving frame as above we get
n+1
n+1
un+1 (y) = Ui−1
Li−1 (y) + Uin+1 Li (y) + Ui+1
Li+1 (y),
(22)
where Ui = exp((ln ûx̂ )d (x̂ − xn+1
))un+1
, as in the case of the invariant linear interpolation (20).
i
i
Numerical examples using the invariant quadratic interpolation will be given in Section 8.
Interpolation using dif ference invariants. The product frame on the grid point space
allows invariantizing the elementary variables xni and uni , which yields the system of joint invariants. In the continuous limit these invariants take the normalization values chosen for x and u
to construct the usual moving frame ρ [27]. On the other hand, on the discrete space Mξn we
only normalize one xni (i, n fixed) among all the grid points xkl and the analog statement is true
for the associated values ukl . This means that the joint invariants ι(xkl ) and ι(ukl ), l 6= i, k 6= n,
are nontrivial and can be used to assemble invariant interpolation schemes.
In the present case, while we have normalized uni = 1 in the course of constructing the moving
frame ρ4 , we are free to use the moving frame to invariantized any ukl where l 6= i, k 6= n and
this will yield a proper (nontrivial) invariant on the discrete space Mξ4 . As above, we again
recompute the moving frame for G1 by replacing the normalization conditions uni = 1 and
udx = 0 with un+1
= 0 and ûdx̂ = 0, respectively, which yields new expressions for the moving
i
frame components of ε3 and ε5 given by
ε3 = −
ln un+1
− xn+1
(ln ûx̂ )d − τ n+1 (ln ux̂ )d
i
i
2 ,
ε5 = (ln ûx̂ )d .
18
A. Bihlo and J.-C. Nave
Using this modified moving frame, we then invarianize the variable un+1 (ξi ), which is the sought
value of u at the point (τ n+1 , ξi ) of the computational domain. This invariantization yields
un+1 (ξi )
d n+1
ι un+1 (ξi ) =
exp
(ln
û
)
x
−
ξ
.
i
x̂
i
un+1
i
Because in the continuous limit the invariantization of un+1 (ξi ) must reproduce the normalization
condition u = 1, we restrict the difference invariant to the manifold ι(un+1 (ξi )) = 1. The
invariant interpolation is thus
n+1
un+1 (ξi ) = exp (ln ûx̂ )d ξi − xn+1
ui
i
and it is again consistent as un+1 (xn+1
) = un+1
. More accurate interpolations could be coni
i
structed by combining the invariants ι(xkl ) and ι(ukl ) in a suitable way.
The advantage of the interpolation methods introduced in this section is that they are invariant under the group G1 , i.e. using these interpolation formulas to map un+1
back to the
i
ξ-grid does not break the invariance of the numerical schemes for the heat equation, while still
allowing one to use a regular grid. Invariant interpolations thus allow avoiding the complications that moving meshes impose on the applicability of symmetry-preserving finite difference
discretizations.
8
Numerical verif ication
In order to verify the accuracy predicted above for the various schemes proposed, we set-up the
following problem. On a periodic domain x ∈ [0, 2π[, consider
ut = uxx ,
u(x, t = 0) = sin(x − 1) + 2.
On a sequence of grids with N ∈ {2, 4, 8, . . . , 256}, the number of grid points, we compute the
error in the maximum norm between the numerical solution and the exact solution at t = 1.
The time step ∆τ is taken as proportional to h2 , h = h+ = h− , in all simulations.
In each of the following figures, we plot the reference line corresponding to O h2 dash–
dotted, and the L∞ error as black line where,
kEkL∞ = max |u(x, 1) − uexact t(x, 1)| .
x∈[0,2π]
Note that for all approaches described below we expect a second order convergence since the
numerical scheme being used is of second order, its invariantization was shown to preserve this
order and the quadratic interpolation and its invariantization is also of second order.
8.1
Invariant scheme without projection
In this test run we use the scheme (11) without projection. As a result, the solution is evolving
along the trajectories of the grid equation (12). Since the spacing between trajectories is not
constant, i.e. h+ and h− are changing in time, we choose to plot the error versus 1/N . In Fig. 2,
we observe the second order convergence expected.
8.2
Invariant scheme with non-invariant quadratic interpolation
In this scheme we interpolate the solution of the invariant scheme (11) at every step back onto
the regular grid using standard Lagrange quadratic interpolation (21). As a result, we have
the solution on a regular grid with step-size h = 1/N . In Fig. 3, we observe the second order
convergence expected.
Invariant Discretization Schemes Using Evolution–Projection Techniques
19
1
10
0
10
−1
∞
|u−uexact|L
10
−2
10
−3
10
−4
10
−5
10
−2
10
−1
0
10
10
1
10
1/N
Figure 2. Convergence plot for the invariant scheme (11) with invariant grid equation (12) and without
re-mapping.
1
10
0
10
−1
∞
|u−uexact|L
10
−2
10
−3
10
−4
10
−5
10
−2
10
−1
0
10
10
1
10
h
Figure 3. Convergence plot for the invariant scheme (11) with invariant grid equation (12) using noninvariant quadratic interpolation (21) as projection.
8.3
Invariant scheme with invariant quadratic interpolation
In this scheme, we interpolate the solution of the invariant scheme (11) at every step back
onto the regular grid using the invariant Lagrange quadratic interpolation (22) described in the
previous section. As for the case above, at each time step we have the solution on a regular grid
with step-size h = 1/N . In Fig. 4, we observe the second order convergence expected.
20
A. Bihlo and J.-C. Nave
1
10
0
10
−1
∞
|u−uexact|L
10
−2
10
−3
10
−4
10
−5
10
−2
10
−1
0
10
10
1
10
h
Figure 4. Convergence plot for the invariant scheme (11) with invariant grid equation (12) using invariant
quadratic interpolation (22) as projection.
8.4
Linearity preservation in the invariant numerical scheme
Linearity is not preserved by construction in the schemes proposed. For this reason it is instructive to check numerically whether or not linearity is preserved in the fully invariant scheme.
Consider the initial value problem,
ut = uxx
u(x, t = 0) = (sin(x − 1) + 2) + (cos x + 2),
the solution of which we call uexact . We then solve numerically the following two equations
uat = uaxx with ua (x, 0) = sin(x − 1) + 2,
ubt = ubxx with ub (x, 0) = cos x + 2,
and define us = ua + ub .
Fig. 5 depicts the L∞ error between uexact and us . We observe a convergence rate of second
order. In other words, despite the fully invariant numerical scheme does not explicitly preserve
the symmetry associated with the linear superposition principle, we observe that the linearity
property is preserved approximately to the order of the method.
9
Conclusions
In this paper we construct invariant discretization schemes using the method of invariantization
via equivariant moving frames. The advantage of this technique is that it allows one to start
with a given non-invariant scheme and convert this initial scheme into a finite difference approximation of a system of differential equations L that is invariant under the same maximal
Lie invariance group G (or a suitably chosen subgroup of G) as admitted by L.
The possibility of converting non-invariant numerical schemes into invariant discretizations
may lead to the overly optimistic speculation that the schemes constructed by invariantization
Invariant Discretization Schemes Using Evolution–Projection Techniques
21
1
10
0
10
−1
∞
|u−uexact|L
10
−2
10
−3
10
−4
10
−5
10
−2
10
−1
0
10
10
1
10
h
Figure 5. Convergence plot for the linearity test using the invariant scheme (11) with invariant grid
equation (12) using invariant quadratic interpolation (22) as projection.
could be easily included in existing numerical models using the original scheme. The hurdle
preventing this in practice is that preserving symmetry groups of systems of evolution equations
more complicated than scalings or translations requires the use of moving grids. Converting
numerical models that use standard discretization schemes based on fixed lattices to (invariant)
discretization schemes on moving meshes is not an easy task. At the same time, rewriting
numerical models from scratch for the simulation of involved physical processes using symmetrypreserving schemes might be a time-consuming and costly task too and it not certain that this
is feasible at all. Moreover, it is as of now unclear whether preserving symmetries in numerical
schemes for multi-dimensional systems of partial differential equations gives enough added value
compared to standard schemes that one might justify such an undertaking in practice.
This is why one relies on finding methods allowing one to efficiently include invariant discretization schemes into existing numerical models without the need to rewrite new models
from scratch that incorporate the invariance methodology. The method proposed in this article
solves this problem by breaking the integration procedure into two steps, the time-stepping using
the invariant numerical scheme with an invariant numerical grid equation and the projection
(interpolation) of the results obtained at intermediate grid points to the regular mesh. This
interpolation can be done in an invariant way by applying the moving frame map used to invariantize the initial discretization scheme also to a particular interpolation method. An alternative
is to assemble the invariant interpolation method using joint invariants. Either way, it is worthwhile pointing out that interpolations requiring only data given at a single time level are already
invariant under most symmetry groups as admitted by physical systems of differential equations.
Thus, invariantization of interpolation formulas will often only lead to minor modifications of
the initial interpolation method chosen and the influence on the numerical solution might be
rather small. In the numerical tests carried out above for the heat equation, the difference in the
convergence properties we found when using invariant or non-invariant interpolation methods
is indeed small although using the invariant interpolation gave slightly better numerical results. This is encouraging and the reason why we plan to further investigate invariant numerical
schemes using the projection procedure.
22
A. Bihlo and J.-C. Nave
We illustrate the evolution–projection strategy by integrating the one-dimensional linear
heat equation with an invariant numerical scheme. The heat equation has been studied quite
extensively in light of its invariance properties and in particular it is a standard model for
the construction of invariant numerical schemes [1, 14, 35]. At the same time, a comprehensive numerical analysis of such schemes was not given before and thus seems relevant to be
reported. This is another aim of the present paper. Again, the analysis of numerical properties
of discretization schemes is considerably easier if one can use non-evolving meshes.
Further work we intend to do is to employ the evolution–projection strategy to multidimensional systems of differential equations using both higher-order discretization and interpolation schemes.
Acknowledgements
The authors thank Professor Roman Popovych for valuable discussions and careful reading of
the manuscript. The valuable remarks of the anonymous referees are much appreciated. This
research was supported by the Austrian Science Fund (FWF), project J3182–N13 (AB). JCN
wishes to acknowledge partial support from the NSERC Discovery Program, and the National
Science Foundation through grant DMS-0813648.
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